Eigenvalue multiplicity of a graph in terms of the number of external vertices (Q6541324)

Let \(m(G, \lambda)\) be the multiplicity of an eigenvalue \(\lambda\) of a graph \(G\). In a connected graph \(G\) with at least two vertices, a vertex is called external if it is not a cut vertex and let \(\epsilon(G)\) be the number of external vertices of \(G\). In this paper, the authors prove that \(m(G, \lambda) \leq \epsilon(G)-1\) for any \(\lambda\) and characterize the extremal graphs with \(m(G,-1) = \epsilon(G)-1\), which generalizes the main result of \textit{X. Wang} et al. [Linear Multilinear Algebra 70, No. 17, 3345--3353 (2022; Zbl 1505.05095)] from a tree to an arbitrary connected graph. An open problem concludes the paper.